The Psychic Power Behind Your System
There’s a tacit connection between magic, forces, and energy; however, energy is an abstract concept. Waves tacitly are understood to have energy, so in thinking about energy, you can think of the energy that waves have. Energy is not very useful in its disordered states. By formalizing a set of axioms from which you can build magical constructs, you are creating an order that corresponds with the energy of the construct. As such, the energy of your magical construct is less diffuse and is intrinsically more useful. You can see how the concepts of the magical system you created relate to ordered forms of energy by looking at the harmonics and the waves of your system.
The structure of a magical construct built in this manner will have a corresponding graph from which you can get information describing waves. The structure of the graph will have what is called an adjacency matrix and a degree matrix. From these two matrices of a graph, you can calculate a Laplacian matrix. From a Laplacian matrix, you can derive a wave equation where the eigenvectors will describe the harmonics and the eigenvalues would describe the frequencies at which these things resonate. Now, you don’t have to do Linear Algebra in order to use it; rather, you just have to tacitly understand that the energy of the system is derived from it being intentional – an arrangement of phenomenological objects. You can call these waves psychic energy, so what powers and resonates with your system is psychic energy which comes from how phenomenological dimensions are differentiated. The adjacency matrix for the example can be seen below:
Magical Decagon Adjacency Matrix
The fun with this is you started off with a basic set of axioms and a basic set of rules and because there is an infinite amount of different ways for the magic to unfold, you can discover new things even though you created the axioms. You can treat these magical archetypes as an alphabet where you can build objects from an alphabet of axioms and rules. There are many different directions you can take this and when you get into the specifics of Linear Algebra and other aspects of Abstract Algebra, you can take it to a whole other level. This article gets you started on the path of doing Thaumaturgy without having to default to occultism.
More than Sigil Magic
It is important to note that this is more abstract than sigil magic. The backbone for how such a magical system is created is through quotients. Unlike the set of all integers, the set of all rational numbers is a mathematical field; therefore, a collection of corresponding objects would be contained within a field. What you are doing is creating a magical field from an abelian group.